| L(s) = 1 | + 2.43·2-s − 2.06·4-s + 1.94·5-s − 13.0·7-s − 24.5·8-s + 4.75·10-s + 53.8·11-s − 82.5·13-s − 31.7·14-s − 43.2·16-s + 17.4·17-s + 67.9·19-s − 4.01·20-s + 131.·22-s + 16.1·23-s − 121.·25-s − 201.·26-s + 26.8·28-s − 14.4·29-s − 146.·31-s + 90.7·32-s + 42.5·34-s − 25.3·35-s − 137.·37-s + 165.·38-s − 47.7·40-s + 475.·41-s + ⋯ |
| L(s) = 1 | + 0.861·2-s − 0.257·4-s + 0.174·5-s − 0.703·7-s − 1.08·8-s + 0.150·10-s + 1.47·11-s − 1.76·13-s − 0.605·14-s − 0.675·16-s + 0.249·17-s + 0.820·19-s − 0.0449·20-s + 1.27·22-s + 0.146·23-s − 0.969·25-s − 1.51·26-s + 0.181·28-s − 0.0926·29-s − 0.847·31-s + 0.501·32-s + 0.214·34-s − 0.122·35-s − 0.609·37-s + 0.707·38-s − 0.188·40-s + 1.81·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.072620393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.072620393\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 2.43T + 8T^{2} \) |
| 5 | \( 1 - 1.94T + 125T^{2} \) |
| 7 | \( 1 + 13.0T + 343T^{2} \) |
| 11 | \( 1 - 53.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 82.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 16.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 14.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 475.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 515.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 458.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 646.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 569.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 599.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 690.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 294.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 329.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 259.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 615.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 441.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.003078960924677460630228083997, −7.80766033710373847323747036234, −6.97254729822984275569548064139, −6.25121112794651575101891386527, −5.44330651488862739590801293446, −4.71626284822174028158128988281, −3.80061942337122841539667389880, −3.17606210218701426095817197805, −2.03527008571424009638295864374, −0.55718987508732116646756298493,
0.55718987508732116646756298493, 2.03527008571424009638295864374, 3.17606210218701426095817197805, 3.80061942337122841539667389880, 4.71626284822174028158128988281, 5.44330651488862739590801293446, 6.25121112794651575101891386527, 6.97254729822984275569548064139, 7.80766033710373847323747036234, 9.003078960924677460630228083997
